1. Field of the Invention
The present invention is generally related to hydrocarbon well stimulation, and is more particularly directed to a method for designing matrix treatment. The invention is particularly useful for designing acid treatment in carbonate reservoirs.
2. Discussion of the Prior Art
Matrix acidizing is a widely used well stimulation technique. The primary objective in this process is to reduce the resistance to the flow of reservoir fluids due to a naturally tight formation or damages. Acid dissolves the material in the matrix and creates flow channels that increase the permeability of the matrix. The efficiency of this process depends on the type of acid used, injection conditions, structure of the medium, fluid to solid mass transfer, reaction rates, etc. While dissolution increases the permeability, the relative increase in the permeability for a given amount of acid is observed to be a strong function of the injection conditions.
In sandstone reservoirs, reaction fronts tend to be uniform and flow channeling is not observed. In carbonate reservoirs, depending on the injection conditions, multiple dissolution patterns may be produced, varying from uniform, conical and wormhole types. At very low flow rates, acid is spent soon after it contacts the medium resulting in face dissolution. The dissolution patterns are observed to be more uniform at high flow rates. At intermediate flow rates, long conductive channels known as wormholes are formed. These channels penetrate deep into the formation and facilitate the flow of oil. Experiments conducted in carbonate cores have shown that the relative increase in permeability for a given amount of acid injected is observed to be higher in wormholes. Thus, for optimizing a stimulation treatment, it is desirable to identify the parameters (e.g: rate of injection, acid type, thickness and permeability of the damaged zone etc.) that will produce wormholes with optimum density and penetrating deep into the formation.
It is well known that the optimum injection rate depends on the reaction and diffusion rates of the acid species, concentration of the acid, length of the core sample, temperature, permeability of the medium etc. The influence of the above factors on the wormhole formation is studied in the experiments. Several theoretical studies have been conducted in the past to obtain an estimate of the optimum injection rate and to understand the phenomena of flow channeling associated with reactive dissolution in porous media. However, the existing models describe only a few aspects of the acidizing process and the coupling of the mechanisms of reaction and transport at various scales that play a key role in the estimation of optimum injection rate are not properly accounted for in these models.
Several models have been proposed that are based on the assumption of an existing wormhole. Reference is made for instance to Wang, Y., Hill, A. D., and Schechter, R. S.:“The Optimum Injection Rate for Matrix Acidizing of Carbonate Formations,” paper SPE 26578 presented at 1993 SPE Annual Technical Conference and Exhibition held in Houston, Tex., Oct. 3-6, 1993; Buijse, M. A.:“Understanding Wormholing Mechanisms Can Improve Acid Treatments in Carbonate Formations,” SPE Prod. & Facilities, 15 (3), 168-175, 2000; and Huang, T., Zhu, D. and Hill, A. D.: “Prediction of Wormhole Population Density in Carbonate Matrix Acidizing,” paper SPE 54723 presented at the 1999 SPE European Formation Damage Conference held in The Hague, May 31-Jun. 1, 1999.
These models are used to study the effect of fluid leakage, reaction kinetics etc., on the wormhole propagation rate and the effect of neighboring wormholes on growth rate of the dominant wormhole. The simple structure of these models offers the advantage of studying the reaction, diffusion and convection mechanisms inside the wormhole in detail. These models, however, cannot be used to study wormhole initiation and the effect of heterogeneities on wormhole formation.
Network models describing reactive dissolution have been presented in Hoefner M. L. and Fogler. H. S.: “Pore Evolution and Channel Formation During Flow and Reaction in Porous Media,” AIChE J, 34, 45-54 (1988); and Fredd, C. N. and Fogler, H. S.: “Influence of Transport and Reaction on Wormhole Formation in Porous Media,” AIChE J, 44, 1933-1949 (1998). These models represent the porous medium as a network of tubes interconnected to each other at the nodes. Acid flow inside these tubes is described using Hagen-Poiseuille relationship for laminar flow inside a pipe. The acid reacts at the wall of the tube and dissolution is accounted in terms of increase in the tube radius. Network models are capable of predicting the dissolution patterns and the qualitative features of dissolution like optimum flow rate, observed in the experiments. However, a core scale simulation of the network model requires huge computational power and incorporating the effects of pore merging and heterogeneities into these models is difficult. The results obtained from network models are also subject to scale up problems.
An intermediate approach to describing reactive dissolution involves the use of averaged or continuum models. Averaged models were used to describe the dissolution of carbonates by Pomès, V., Bazin, B., Golfier, F., Zarcone, C., Lenormand, R. and Quintard, M.: “On the Use of Upscaling Methods to Describe Acid Injection in Carbonates,” paper SPE 71511 presented at 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, La., September 30-Oct. 3, 2001; and Golfier, F., Bazin, B., Zarcone, C., Lenormand, R., Lasseux, D. and Quintard, M.: “On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium,” J. Fluid Mech., 457, 213-254 (2002). Unlike the network models that describe dissolution from the pore scale and the models based on the assumption of existing wormholes, the averaged models describe dissolution at a scale much larger than the pore scale and much smaller than the scale of the core. This intermediate scale is also known as the Darcy scale.
Averaged models circumvent the scale-up problems associated with network models, can predict wormhole initiation, propagation and can be used to study the effects of heterogeneities in the medium on the dissolution process. The results obtained from the averaged models can be extended to the field scale. The success of these models depends on the key inputs such as mass transfer rates, permeability-porosity correlation etc., which depend on the processes that occur at the pore scale. The averaged model written at the Darcy scale requires these inputs from the pore scale. Since the structure of the porous medium evolves with time, a pore level calculation has to be made at each stage to generate inputs for the averaged equation.
Averaged equations used by Golfier et al. and Pomès et al. describe the transport of the reactant at the Darcy scale with a pseudo-homogeneous model, i.e., they use a single concentration variable. In addition, they assume that the reaction is mass transfer controlled (i.e. the reactant concentration at the solid-fluid interface is zero).
The inventors have found that most systems fall in between the mass transfer and kinetically controlled regimes of reaction where the use of a pseudo-homogeneous model (single concentration variable) is not sufficient to capture all the features of the reactive dissolution process qualitatively and that ‘a priori’ assumption that the system is in the mass transfer controlled regime, often made in the literature, may not retain the qualitative features of the problem.
It would be therefore desirable to provide an improved model for predicting the dissolution pattern during matrix stimulation of carbonates.